Spinors
This article is Graduate level Spinors :See also: External link:An introduction to spinors Wikipedia:Rotor (mathematics) in three-dimensions are quaternions}}, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.Wikipedia:Spinor#Three_dimensions .Wikipedia:Spinor Back to top Spinors in three dimensions :From Wikipedia:Spinors in three dimensions The association of a spinor with a 2×2 complex was formulated by Élie Cartan. In detail, : Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space: * det ''X = – (length '''x)2. * X'' 2 = (length '''x')2''I'', where I'' is the identity matrix. * \frac{1}{2}(XY+YX)=({\bold x}\cdot{\bold y})I * \frac{1}{2}(XY-YX)=iZ where ''Z is the matrix associated to the cross product z''' = '''x × y'''. * If '''u is a unit vector, then −''UXU'' is the matrix associated to the vector obtained from x''' by reflection in the plane orthogonal to '''u. * It is an elementary fact from that any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R'' is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector '''u'1 followed by the plane perpendicular to u'''2, then Column vectors Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the ) play. Provisionally, a '''spinor' is a column vector : \xi=\left\begin{matrix}\xi_1\\\xi_2\end{matrix}\right, with complex entries ξ''1 and ''ξ''2. The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors. Often, the first example of spinors that a student of physics encounters are the 2×1 spinors used in Pauli's theory of electron spin. The are a vector of three 2×2 that are used as . in 3 dimensions, for example (''a, b'', ''c), one takes a with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.}} of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.}} Example: u'' = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix: : S_u = (0.8,-0.6,0.0)\cdot \vec{\sigma}=0.8 \sigma_{1}-0.6\sigma_{2}+0.0\sigma_{3} = \begin{bmatrix} 0.0 & 0.8+0.6i \\ 0.8-0.6i & 0.0 \end{bmatrix} The eigenvectors may be found by the usual methods of , but a convenient trick is to note that a Pauli spin matrix is an , that is, the squareof the above matrix is the identity matrix. Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± ''Su. That is, : S_u (1\pm S_u) = \pm 1 (1 \pm S_u) One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are: : \begin{bmatrix} 1.0+ (0.0)\\ 0.0 +(0.8-0.6i) \end{bmatrix}, \begin{bmatrix} 1.0- (0.0)\\ 0.0-(0.8-0.6i) \end{bmatrix} The trick used to find the eigenvectors is related to the concept of , that is, or and therefore each generates an ideal in the Pauli algebra.}} The same trick works in any , in particular the that are discussed below. These projection operators are also seen in theory where they are examples of pure density matrices. : While the two columns appear different, one can use ''a''2 + ''b''2 + ''c''2 = 1 to show that they are multiples (possibly zero) of the same spinor. Back to top Column vectors :From Wikipedia:Spinor: (Note: \nu is the number of an object can have in n dimensions.) Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors). Back to top Electron spin :From Wikipedia:Spinor: . This is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists. Back to top Clifford group From Wikipedia:Clifford algebra The class of Clifford groups ( Clifford–Lipschitz groups ) was discovered by . In this section we assume that V'' is finite-dimensional and the quadratic form ''Q is . An action on the elements of a Clifford algebra by its may be defined in terms of a twisted conjugation: twisted conjugation by x'' maps , where ''α is the main involution defined . The Clifford group Γ is defined to be the set of invertible elements x'' that ''stabilize the set of vectors under this action, meaning that for all v'' in ''V we have: : \alpha(x) v x^{-1}\in V . This formula also defines an action of the Clifford group on the vector space V'' that preserves the quadratic form ''Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r'' of ''V for which Q''(''r) is invertible in K'', and these act on ''V by the corresponding reflections that take v'' to . (In characteristic 2 these are called orthogonal transvections rather than reflections.) If ''V is a finite-dimensional real vector space with a quadratic form then the Clifford group maps onto the orthogonal group of V'' with respect to the form (by the ) and the kernel consists of the nonzero elements of the field ''K. This leads to exact sequences : 1 \rightarrow K^* \rightarrow \Gamma \rightarrow \mbox{O}_V(K) \rightarrow 1,\, : 1 \rightarrow K^* \rightarrow \Gamma^0 \rightarrow \mbox{SO}_V(K) \rightarrow 1.\, Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm. Back to top Spinor norm From Wikipedia:Clifford algebra In arbitrary characteristic, the Q'' is defined on the Clifford group by : Q(x) = x^\mathrm{t}x. It is a homomorphism from the Clifford group to the group ''K× of non-zero elements of K''. It coincides with the quadratic form ''Q of V'' when ''V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ1. The difference is not very important in characteristic other than 2. The nonzero elements of K'' have spinor norm in the group (''K×)2 of squares of nonzero elements of the field K''. So when ''V is finite-dimensional and non-singular we get an induced map from the orthogonal group of V'' to the group ''K×/(K''×)2, also called the spinor norm. The spinor norm of the reflection about ''r⊥, for any vector r'', has image ''Q(r'') in ''K×/(K''×)2, and this property uniquely defines it on the orthogonal group. This gives exact sequences: : 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to \mbox{O}_V(K) \to K^{\times}/(K^{\times})^2, : 1 \to \{\pm 1\} \to \mbox{Spin}_V(K) \to \mbox{SO}_V(K) \to K^{\times}/(K^{\times})^2. Note that in characteristic 2 the group {±1} has just one element. From the point of view of of s, the spinor norm is a on cohomology. Writing μ2 for the (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence : 1 \to \mu_2 \rightarrow \mbox{Pin}_V \rightarrow \mbox{O}_V \rightarrow 1 yields a long exact sequence on cohomology, which begins : 1 \to H^0(\mu_2;K) \to H^0(\mbox{Pin}_V;K) \to H^0(\mbox{O}_V;K) \to H^1(\mu_2;K). The 0th Galois cohomology group of an algebraic group with coefficients in ''K is just the group of K''-valued points: , and , which recovers the previous sequence : 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to \mbox{O}_V(K) \to K^{\times}/(K^{\times})^2, where the spinor norm is the connecting homomorphism . Back to top Spin and Pin groups From Wikipedia:Clifford algebra In this section we assume that V'' is finite-dimensional and its bilinear form is non-singular. (If ''K has characteristic 2 this implies that the dimension of V'' is even.) The Pin''V(K'') is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin''V(K'') is the subgroup of elements of 0 in Pin''V(K''). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group. Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the to be the image of Γ0. If ''K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K'' does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0. There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm . The kernel consists of the elements +1 and −1, and has order 2 unless ''K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V''. In the common case when ''V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V'' has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin(''n), is a double cover of SO(n''). Please note, however, that the simple connectedness of the spin group is not true in general: if ''V is R''p'',q'' for ''p and q'' both at least 2 then the spin group is not simply connected. In this case the algebraic group Spin''p,q'' is simply connected as an algebraic group, even though its group of real valued points Spin''p,q''('R') is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups. Back to top Spinors From Wikipedia:Clifford algebra Clifford algebras Cℓ ('C'), with even, are matrix algebras which have a complex representation of dimension 2''n''. By restricting to the group Pin''p'',q''('R') we get a complex representation of the Pin group of the same dimension, called the . If we restrict this to the spin group Spin''p,q''('R') then it splits as the sum of two ''half spin representations (or Weyl representations) of dimension 2''n''−1. If is odd then the Clifford algebra Cℓ ('C') is a sum of two matrix algebras, each of which has a representation of dimension 2''n, and these are also both representations of the Pin group Pin''p'',q''('R'). On restriction to the spin group Spin''p,q''('R') these become isomorphic, so the spin group has a complex spinor representation of dimension 2''n. More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the : whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on s. Back to top Real spinors From Wikipedia:Clifford algebra To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The , Pin''p'',q'' is the set of invertible elements in Cℓ that can be written as a product of unit vectors: : {\mbox{Pin}}_{p,q}=\{v_1v_2\dots v_r |\,\, \forall i\, \|v_i\|=\pm 1\}. Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group . The consists of those elements of Pin''p, q'' which are products of an even number of unit vectors. Thus by the Spin is a cover of the group of proper rotations . Let be the automorphism which is given by the mapping acting on pure vectors. Then in particular, Spin''p'',q'' is the subgroup of Pin''p,q'' whose elements are fixed by ''α. Let : \operatorname{C\ell}_{p,q}^{0} = \{ x\in \operatorname{C\ell}_{p,q} |\, \alpha(x)=x\}. (These are precisely the elements of even degree in Cℓ .) Then the spin group lies within Cℓ . The irreducible representations of Cℓ restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Cℓ . To classify the pin representations, one need only appeal to the . To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above) : \operatorname{C\ell}^{0}_{p,q} \approx \operatorname{C\ell}_{p,q-1}, \text{ for } q > 0 : \operatorname{C\ell}^{0}_{p,q} \approx \operatorname{C\ell}_{q,p-1}, \text{ for } p > 0 and realize a spin representation in signature as a pin representation in either signature or . Back to top The Lie derivative of a spinor field From Wikipedia:Lie derivative A definition for Lie derivatives of along generic spacetime vector fields, not necessarily ones, on a general (pseudo) was already proposed in 1972 by . Later, it was provided a geometric framework which justifies her ''ad hoc prescription within the general framework of Lie derivatives on in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories. In a given , that is in a Riemannian manifold (M,g) admitting a , the Lie derivative of a \psi can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the 's local expression given in 1963: : \mathcal{L}_X \psi := X^{a}\nabla_{a}\psi -\frac14\nabla_{a}X_{b} \gamma^{a}\,\gamma^{b}\psi\, , where \nabla_{a}X_{b}=\nabla_{a}X_{b} , as X=X^{a}\partial_{a} is assumed to be a , and \gamma^{a} are . It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a generic vector field X , but explicitly taking the antisymmetric part of \nabla_{a}X_{b} only. More explicitly, Kosmann's local expression given in 1972 is: : \mathcal{L}_X \psi := X^{a}\nabla_{a}\psi -\frac18\nabla_{a}X_{b} \gamma^{a},\gamma^{b}\psi\, = \nabla_X \psi - \frac14 (d X^\flat)\cdot \psi\, , where \gamma^{a},\gamma^{b}= \gamma^a\gamma^b - \gamma^b\gamma^a is the commutator, d is , X^\flat = g(X, -) is the dual 1 form corresponding to X under the metric (i.e. with lowered indices) and \cdot is Clifford multiplication. It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the . This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the . Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel. To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article, where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the . Back to top Quantum mechanics In , each Pauli matrix is related to an that corresponds to an describing the of a particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, are the generators of a (spin representation) of the acting on particles with spin ½. The of the particles are represented as two-component . In the same way, the Pauli matrices are related to the . An interesting property of spin ½ particles is that they must be rotated by an angle of 4 in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2, they are actually represented by vectors in the two dimensional complex . For a spin ½ particle, the spin operator is given by σ}}, the of . By taking s of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting s for higher spin systems in three spatial dimensions, for arbitrarily large ''j, can be calculated using this and . They can be found in . The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. Also useful in the of multiparticle systems, the general is defined to consist of all -fold products of Pauli matrices. Back to top Pauli matrices Relativistic quantum mechanics From Wikipedia:Pauli matrices In , the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as : \mathsf{\Sigma}_i = \begin{pmatrix} \mathsf{\sigma}_i & 0 \\ 0 & \mathsf{\sigma}_i \end{pmatrix}. It follows from this definition that \mathsf{\Sigma}_i matrices have the same algebraic properties as \mathsf{\sigma}_i matrices. However, is not a three-vector, but a second order . Hence \mathsf{\Sigma}_i needs to be replaced by \Sigma_{\mu\nu} , the generator of . By the antisymmetry of angular momentum, the \Sigma_{\mu\nu} are also antisymmetric. Hence there are only six independent matrices. The first three are the \Sigma_{jk}\equiv \epsilon_{ijk}\mathsf{\Sigma}_i . The remaining three, -i\Sigma_{0i}\equiv\mathsf{\alpha}_i , where the are defined as : \mathsf{\alpha}_i = \begin{pmatrix} 0 & \mathsf{\sigma}_i\\ \mathsf{\sigma}_i & 0\end{pmatrix}. The relativistic spin matrices \Sigma_{\mu\nu} are written in compact form in terms of commutator of as : \Sigma_{\mu\nu} = \frac{i}{2}\left\gamma_\nu\right . Back to top Quantum information From Wikipedia:Pauli matrices In , single- s are 2'' × ''2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the Z–Y decomposition of a single-qubit gate. Choosing a different Cartan pair gives a similar X–Y decomposition of a single-qubit gate. Back to top References Category:Spinors